Properties

Label 7098.2.a.by
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + ( -1 + \beta ) q^{5} + q^{6} + q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + ( -1 + \beta ) q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + ( -1 + \beta ) q^{10} -3 \beta q^{11} + q^{12} + q^{14} + ( -1 + \beta ) q^{15} + q^{16} + ( -4 + \beta ) q^{17} + q^{18} + ( -3 + 2 \beta ) q^{19} + ( -1 + \beta ) q^{20} + q^{21} -3 \beta q^{22} + ( -4 - 2 \beta ) q^{23} + q^{24} + ( -1 - 2 \beta ) q^{25} + q^{27} + q^{28} + ( -7 + 2 \beta ) q^{29} + ( -1 + \beta ) q^{30} + ( -5 + \beta ) q^{31} + q^{32} -3 \beta q^{33} + ( -4 + \beta ) q^{34} + ( -1 + \beta ) q^{35} + q^{36} + ( -5 - \beta ) q^{37} + ( -3 + 2 \beta ) q^{38} + ( -1 + \beta ) q^{40} + ( -5 - 2 \beta ) q^{41} + q^{42} + ( -7 - \beta ) q^{43} -3 \beta q^{44} + ( -1 + \beta ) q^{45} + ( -4 - 2 \beta ) q^{46} + ( 3 - 4 \beta ) q^{47} + q^{48} + q^{49} + ( -1 - 2 \beta ) q^{50} + ( -4 + \beta ) q^{51} + ( -3 - 4 \beta ) q^{53} + q^{54} + ( -9 + 3 \beta ) q^{55} + q^{56} + ( -3 + 2 \beta ) q^{57} + ( -7 + 2 \beta ) q^{58} + ( 2 + 4 \beta ) q^{59} + ( -1 + \beta ) q^{60} + ( 2 + \beta ) q^{61} + ( -5 + \beta ) q^{62} + q^{63} + q^{64} -3 \beta q^{66} + 10 q^{67} + ( -4 + \beta ) q^{68} + ( -4 - 2 \beta ) q^{69} + ( -1 + \beta ) q^{70} + ( -9 + \beta ) q^{71} + q^{72} + ( -2 + 2 \beta ) q^{73} + ( -5 - \beta ) q^{74} + ( -1 - 2 \beta ) q^{75} + ( -3 + 2 \beta ) q^{76} -3 \beta q^{77} + 9 q^{79} + ( -1 + \beta ) q^{80} + q^{81} + ( -5 - 2 \beta ) q^{82} + ( 11 - \beta ) q^{83} + q^{84} + ( 7 - 5 \beta ) q^{85} + ( -7 - \beta ) q^{86} + ( -7 + 2 \beta ) q^{87} -3 \beta q^{88} + ( 3 + 8 \beta ) q^{89} + ( -1 + \beta ) q^{90} + ( -4 - 2 \beta ) q^{92} + ( -5 + \beta ) q^{93} + ( 3 - 4 \beta ) q^{94} + ( 9 - 5 \beta ) q^{95} + q^{96} + ( 9 + 3 \beta ) q^{97} + q^{98} -3 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} - 2q^{10} + 2q^{12} + 2q^{14} - 2q^{15} + 2q^{16} - 8q^{17} + 2q^{18} - 6q^{19} - 2q^{20} + 2q^{21} - 8q^{23} + 2q^{24} - 2q^{25} + 2q^{27} + 2q^{28} - 14q^{29} - 2q^{30} - 10q^{31} + 2q^{32} - 8q^{34} - 2q^{35} + 2q^{36} - 10q^{37} - 6q^{38} - 2q^{40} - 10q^{41} + 2q^{42} - 14q^{43} - 2q^{45} - 8q^{46} + 6q^{47} + 2q^{48} + 2q^{49} - 2q^{50} - 8q^{51} - 6q^{53} + 2q^{54} - 18q^{55} + 2q^{56} - 6q^{57} - 14q^{58} + 4q^{59} - 2q^{60} + 4q^{61} - 10q^{62} + 2q^{63} + 2q^{64} + 20q^{67} - 8q^{68} - 8q^{69} - 2q^{70} - 18q^{71} + 2q^{72} - 4q^{73} - 10q^{74} - 2q^{75} - 6q^{76} + 18q^{79} - 2q^{80} + 2q^{81} - 10q^{82} + 22q^{83} + 2q^{84} + 14q^{85} - 14q^{86} - 14q^{87} + 6q^{89} - 2q^{90} - 8q^{92} - 10q^{93} + 6q^{94} + 18q^{95} + 2q^{96} + 18q^{97} + 2q^{98} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
1.00000 1.00000 1.00000 −2.73205 1.00000 1.00000 1.00000 1.00000 −2.73205
1.2 1.00000 1.00000 1.00000 0.732051 1.00000 1.00000 1.00000 1.00000 0.732051
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7098.2.a.by 2
13.b even 2 1 7098.2.a.bo 2
13.f odd 12 2 546.2.s.b 4
39.k even 12 2 1638.2.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.b 4 13.f odd 12 2
1638.2.bj.a 4 39.k even 12 2
7098.2.a.bo 2 13.b even 2 1
7098.2.a.by 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{2} + 2 T_{5} - 2 \)
\( T_{11}^{2} - 27 \)
\( T_{17}^{2} + 8 T_{17} + 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -2 + 2 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -27 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 13 + 8 T + T^{2} \)
$19$ \( -3 + 6 T + T^{2} \)
$23$ \( 4 + 8 T + T^{2} \)
$29$ \( 37 + 14 T + T^{2} \)
$31$ \( 22 + 10 T + T^{2} \)
$37$ \( 22 + 10 T + T^{2} \)
$41$ \( 13 + 10 T + T^{2} \)
$43$ \( 46 + 14 T + T^{2} \)
$47$ \( -39 - 6 T + T^{2} \)
$53$ \( -39 + 6 T + T^{2} \)
$59$ \( -44 - 4 T + T^{2} \)
$61$ \( 1 - 4 T + T^{2} \)
$67$ \( ( -10 + T )^{2} \)
$71$ \( 78 + 18 T + T^{2} \)
$73$ \( -8 + 4 T + T^{2} \)
$79$ \( ( -9 + T )^{2} \)
$83$ \( 118 - 22 T + T^{2} \)
$89$ \( -183 - 6 T + T^{2} \)
$97$ \( 54 - 18 T + T^{2} \)
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